Calculus of Variations and Geometric Measure Theory

G. B. Maggiani - R. Scala - N. Van Goethem

A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity

created by scala on 09 Mar 2014
modified by maggiani on 15 Mar 2017


Mathematical Methods in Applied Sciences

Inserted: 9 mar 2014
Last Updated: 15 mar 2017

Year: 2014


We prove the Saint-Venant compatibility conditions in $L^p$ in a simply connected domain, with $p \in (1,+\infty)$. Moreover we use the Helhmoltz decomposition to deduce that every symmetric $L^p$ tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of the gradient of a displacement, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. We observe that if the displacement satisfies a Dirichlet condition on the boundary, the decomposition is unique. We apply these results to provide an alternative proof of some classical Korn inequalities, which are very important in elasticity